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A ray through the origin intercepts the hyperbola $\scriptstyle x^2\ -\ y^2\ =\ 1$ in the point $\scriptstyle (\cosh\,a,\,\sinh\,a)$, where $\scriptstyle a$ is twice the area between the ray and the $\scriptstyle x$-axis. For points on the hyperbola below the $\scriptstyle x$-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" (typically pronounced /ˈsɪntʃ/ or /ˈʃaɪn/), and the hyperbolic cosine "cosh" (typically pronounced /ˈkɒʃ/), from which are derived the hyperbolic tangent "tanh" (typically pronounced /ˈtæntʃ/ or /ˈθæn/), etc., in analogy to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh", or sometimes by the misnomer of "arcsinh"[1]) and so on.

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, and Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take real values for real argument called a hyperbolic angle. In complex analysis, they are simply rational functions of exponentials, and so are meromorphic.

Hyperbolic functions were introduced in the 18th century by the Swiss mathematician Johann Heinrich Lambert.

Standard algebraic expressions

sinh, cosh and tanh
csch, sech and coth

The hyperbolic functions are:

• Hyperbolic sine:
$\sinh x = \frac{e^x - e^{-x}}{2}$
• Hyperbolic cosine:
$\cosh x = \frac{e^{x} + e^{-x}}{2}$
• Hyperbolic tangent:
$\tanh x = \frac{\sinh x}{\cosh x} = \frac {\frac{1}{2}(e^x - e^{-x})} {\frac{1}{2}(e^x + e^{-x})} = \frac{e^{2x} - 1} {e^{2x} + 1}$
• Hyperbolic cotangent:
$\coth x = \frac{\cosh x}{\sinh x} = \frac {\frac{1}{2}(e^x + e^{-x})} {\frac{1}{2}(e^x - e^{-x})} = \frac{e^{2x} + 1} {e^{2x} - 1}$
• Hyperbolic secant:
$\operatorname{sech}\,x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}$
• Hyperbolic cosecant:
$\operatorname{csch}\,x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}}$

Via complex numbers the hyperbolic functions are related to the circular functions as follows:

• Hyperbolic sine:
$\sinh x = - {\rm{i}} \sin {\rm{i}}x \!$
• Hyperbolic cosine:
$\cosh x = \cos {\rm{i}}x \!$
• Hyperbolic tangent:
$\tanh x = -{\rm{i}} \tan {\rm{i}}x \!$
• Hyperbolic cotangent:
$\coth x = {\rm{i}} \cot {\rm{i}}x \!$
• Hyperbolic secant:
$\operatorname{sech}\,x = \sec { {\rm{i}} x} \!$
• Hyperbolic cosecant:
$\operatorname{csch}\,x = {\rm{i}}\,\csc\,{\rm{i}}x \!$

where ${\rm{i}} \,$ is the imaginary unit defined as ${\rm{i}} ^2=-1\,$.

The complex forms in the definitions above derive from Euler's formula.

Note that, by convention, sinh2x means (sinhx)2, not sinh(sinhx); similarly for the other hyperbolic functions when used with positive exponents. Another notation for the hyperbolic cotangent function is $\operatorname{ctnh}\,x$, though cothx is far more common.

Useful relations

$\sinh(-x) = -\sinh x\,\!$
$\cosh(-x) = \cosh x\,\!$

Hence:

$\tanh(-x) = -\tanh x\,\!$
$\coth(-x) = -\coth x\,\!$
$\operatorname{sech}(-x) = \operatorname{sech}\, x\,\!$
$\operatorname{csch}(-x) = -\operatorname{csch}\, x\,\!$

It can be seen that ${\rm{cosh}}\,x\,$ and ${\rm{sech}}\,x\,$ are even functions; the others are odd functions.

$\operatorname{arsech}\,x=\operatorname{arcosh} \frac{1}{x}$
$\operatorname{arcsch}\,x=\operatorname{arsinh} \frac{1}{x}$
$\operatorname{arcoth}\,x=\operatorname{artanh} \frac{1}{x}$

Hyperbolic sine and cosine satisfy the identity

$\cosh^2 x - \sinh^2 x = 1\,$

which is similar to the Pythagorean trigonometric identity.

The hyperbolic tangent is the solution to the nonlinear boundary value problem[2]:

$\frac 1 2 f'' = f^3 - f \qquad ; \qquad f(0) = f'(\infty) = 0$

It can also be shown that the area under the graph of cosh x from A to B is equal to the arc length of cosh x from A to B.

Inverse functions as logarithms

$\operatorname {arsinh} \, x=\ln \left( x+\sqrt{x^{2}+1} \right)$
$\operatorname {arcosh} \, x=\ln \left( x+\sqrt{x^{2}-1} \right);x\ge 1$
$\operatorname {artanh} \, x=\tfrac{1}{2}\ln \frac{1+x}{1-x} ;\left| x \right|<1$
$\operatorname {arsech} \, x=\ln \frac{1+\sqrt{1-x^{2}}}{x} ;0
$\operatorname {arcsch} \, x=\ln \left( \frac{1}{x}+\frac{\sqrt{1+x^{2}}}{\left| x \right|} \right)$
$\operatorname {arcoth} \, x=\tfrac{1}{2}\ln \frac{x+1}{x-1} ;\left| x \right|>1$

Derivatives

$\frac{d}{dx}\sinh x = \cosh x \,$
$\frac{d}{dx}\cosh x = \sinh x \,$
$\frac{d}{dx}\tanh x = 1 - \tanh^2 x = \hbox{sech}^2 x = 1/\cosh^2 x \,$
$\frac{d}{dx}\coth x = 1 - \coth^2 x = -\hbox{csch}^2 x = -1/\sinh^2 x \,$
$\frac{d}{dx}\ \hbox{csch}\,x = - \coth x \ \hbox{csch}\,x \,$
$\frac{d}{dx}\ \hbox{sech}\,x = - \tanh x \ \hbox{sech}\,x \,$
$\frac{d}{dx}\left( \sinh^{-1}x \right)=\frac{1}{\sqrt{x^{2}+1}}$
$\frac{d}{dx}\left( \cosh^{-1}x \right)=\frac{1}{\sqrt{x^{2}-1}}$
$\frac{d}{dx}\left( \tanh^{-1}x \right)=\frac{1}{1-x^{2}}$
$\frac{d}{dx}\left( \operatorname{csch}^{-1}x \right)=-\frac{1}{\left| x \right|\sqrt{1+x^{2}}}$
$\frac{d}{dx}\left( \operatorname{sech}^{-1}x \right)=-\frac{1}{x\sqrt{1-x^{2}}}$
$\frac{d}{dx}\left( \coth ^{-1}x \right)=\frac{1}{1-x^{2}}$

Standard Integrals

For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions

$\int\sinh ax\,dx = \frac{1}{a}\cosh ax + C$
$\int\cosh ax\,dx = \frac{1}{a}\sinh ax + C$
$\int \tanh ax\,dx = \frac{1}{a}\ln(\cosh ax) + C$
$\int \coth ax\,dx = \frac{1}{a}\ln(\sinh ax) + C$
$\int{\frac{du}{\sqrt{a^{2}+u^{2}}}}=\sinh ^{-1}\left( \frac{u}{a} \right)+C$
$\int{\frac{du}{\sqrt{u^{2}-a^{2}}}}=\cosh ^{-1}\left( \frac{u}{a} \right)+C$
$\int{\frac{du}{a^{2}-u^{2}}}=\frac{1}{a}\tanh ^{-1}\left( \frac{u}{a} \right)+C; u^{2}
$\int{\frac{du}{a^{2}-u^{2}}}=\frac{1}{a}\coth ^{-1}\left( \frac{u}{a} \right)+C; u^{2}>a^{2}$
$\int{\frac{du}{u\sqrt{a^{2}-u^{2}}}}=-\frac{1}{a}\operatorname{sech}^{-1}\left( \frac{u}{a} \right)+C$
$\int{\frac{du}{u\sqrt{a^{2}+u^{2}}}}=-\frac{1}{a}\operatorname{csch}^{-1}\left| \frac{u}{a} \right|+C$

In the above expressions, C is called the constant of integration.

Taylor series expressions

It is possible to express the above functions as Taylor series:

$\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} +\cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$
$\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$
$\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2}$
$\coth x = \frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi$ (Laurent series)
$\operatorname {sech}\, x = 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2}$
$\operatorname {csch}\, x = \frac {1} {x} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi$ (Laurent series)

where

$B_n \,$ is the nth Bernoulli number
$E_n \,$ is the nth Euler number

Similarities to circular trigonometric functions

A point on the hyperbola xy = 1 with x > 1 determines a hyperbolic triangle in which the side adjacent to the hyperbolic angle is associated with cosh while the side opposite is associated with sinh. However, since the point (1,1) on this hyperbola is a distance √2 from the origin, the normalization constant 1/√2 is necessary to define cosh and sinh by the lengths of the sides of the hyperbolic triangle.

Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola x2 − y2 = 1. This is based on the easily verified identity

$\cosh^2 t - \sinh^2 t = 1 \,$

and the property that cosh t ≥ 1 for all t.

The hyperbolic functions are periodic with complex period i (πi for hyperbolic tangent and cotangent).

The parameter t is not a circular angle, but rather a hyperbolic angle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola.

The function cosh x is an even function, that is symmetric with respect to the y-axis.

The function sinh x is an odd function, that is −sinh x = sinh(−x), and sinh 0 = 0.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule [3] states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems

$\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \,$
$\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \,$
$\tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \,$

the "double angle formulas"

$\sinh 2x\ = 2\sinh x \cosh x \,$
$\cosh 2x\ = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1 \,$

and the "half-angle formulas"

$\cosh^2 \tfrac{1}{2} x = \tfrac{1}{2}(\cosh x + 1)$    Note: This corresponds to its circular counterpart.
$\sinh^2 \tfrac{1}{2} x = \tfrac{1}{2}(\cosh x - 1)$    Note: This is equivalent to its circular counterpart multiplied by −1.
$\tanh ^{2}x=1-\operatorname{sech}^{2}x$
$\coth ^{2}x=1+\operatorname{csch}^{2}x$

The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin x).

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain hanging freely under gravity.

Relationship to the exponential function

From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

$e^x = \cosh x + \sinh x\!$

and

$e^{-x} = \cosh x - \sinh x.\!$

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums of complex exponentials.

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic.

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

$e^{i x} = \cos x + i \;\sin x$
$e^{-i x} = \cos x - i \;\sin x$

so:

$\cosh ix = \tfrac12(e^{i x} + e^{-i x}) = \cos x$
$\sinh ix = \tfrac12(e^{i x} - e^{-i x}) = i \sin x$
$\tanh ix = i \tan x \,$
$\cosh x = \cos ix \,$
$\sinh x = -i \sin ix \,$
$\tanh x = -i \tan ix \,$
 $\operatorname{sinh}(z)$ $\operatorname{cosh}(z)$ $\operatorname{tanh}(z)$ $\operatorname{coth}(z)$ $\operatorname{sech}(z)$ $\operatorname{csch}(z)$

References

1. ^ Some examples of using arcsinh found in Google Books.
2. ^ Eric W. Weisstein. "Hyperbolic Tangent". MathWorld. Retrieved 2008-10-20.
3. ^ G. Osborn, Mnemonic for hyperbolic formulae, The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902

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